Binomial Distribution Example_using generating function

Example) 

Let Y~be binomial (15, $\frac {1}{3}$). Evaluate Var(Y).  

(We know variance of binomial distribution is npq. However what if we don't know this formula?) 

$\triangleright$ Think First 

$Var(Y)=E(Y^2)-[E(Y)]^2$. So we need the second moment, which means we need to use generating function. 

$\triangleright$ Solution

By using generating function, $G(z)=E(z^Y)=(q+pz)^n$

$G'(z)=E(Y \cdot z^{Y-1})=n \cdot (q+pz)^{n-1}\cdot p$ 

So when z=1, $G'(1)=E(Y)=np$ 

$G''(z)=E(Y (Y-1) \cdot z^{Y-2})=n (n-1)\cdot (q+pz)^{n-2}\cdot p^2$, 

so when z=1 $G''(1)=E(Y^2-Y)=n(n-1)^2p=E(Y^2)-E(Y)=n(n-1)p^2$  

$Var(Y)=n(n-1)p^2+np-(np)^2=n^2p^2-np^2+np-n^2p^2=npq \because(q=1-p)$ 

$\therefore Var(Y)=15 \cdot \frac{1}{3} \cdot \frac{2}{3}=\frac {30}{9}$